 
Summary: RESEARCH BLOG 10/27/03
I'll make a couple of more comments on the results of Kronheimer et.
al.. Last blog, I discussed 4manifolds constructed from 4 or fewer han
dles. For and introduction to handle decompositions of 4manifolds, see
Gompf and Stipicz's book. The trickiest ones to classify are manifolds
with two 2handles. One way to view this is by considering pushing
one 2handle above the other. The level set in between the 2handles
is a 3manifold, which is obtained by surgery on a knot in S3
in two
different ways, by adding a two handle to the 0handle, or by adding
the other 2handle to the 4handle. The manifold then is determined
by two knots in S3
which have the same surgery. If one performs inte
gral nsurgery on a knot, the resulting manifold has homology Z/nZ.
Thus, to obtain the same manifold, we must be performing the same
surgery (up to orientation reversal) on the two knots. Now, the main
result of Kronheimer et. al. is that if a knot has the same Dehn surgery
as the unknot, then it must be the unknot. This means that if one of
our handles is unknotted, then the other is also. There is still some
ambiguity in forming the manifold, since we must glue the boundary
